Document Zbl 07987126

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Appl. Math.</a> <a href="/?q=in%3A526926" title="Articles in this Issue">44, No. 3, Paper No. 124, 14 p. (2025)</a>. </div> <div class="abstract">Summary: We study a Volterra convolution equation of the second kind, based on a combination of Riemann-Liouville integrals. The problem can be reformulated involving the Caputo fractional derivative, hence the equation becomes of differintegral type. The modeling interpretation is based on a non-Markovian state function, where the Riemann-Liouville multi-orders are memory coefficients that decrease hazard risks of change. We prove the validity of the reformulations with fractional-calculus theory, local existence with fixed-point tools, and global uniqueness with a Gronwall-type argumentation. We show some examples and their associated physics. We also solve the general linear equation by means of the algebraic formalism of Mikusiński operational calculus, which is superior to Laplace transforms or Picard’s iterations. Multivariate Mittag-Leffler functions play a key role. We relate the emerging closed-form solution with the fractional power series that one may expect for these types of models.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A45D05" title="MSC2020">45D05</a> </td> <td class="space"> Volterra integral equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34A08" title="MSC2020">34A08</a> </td> <td class="space"> Fractional ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A33E12" title="MSC2020">33E12</a> </td> <td class="space"> Mittag-Leffler functions and generalizations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AVolterra+integral+equation">Volterra integral equation</a>; <a href="/?q=ut%3Afractional+calculus">fractional calculus</a>; <a href="/?q=ut%3Awell-posedness">well-posedness</a>; <a href="/?q=ut%3Amodeling+interpretation">modeling interpretation</a>; <a href="/?q=ut%3Aclosed-form+solution">closed-form solution</a>; <a href="/?q=ut%3AMikusi%C5%84ski+algebraic+method">Mikusiński algebraic method</a>; <a href="/?q=ut%3Amultivariate+Mittag-Leffler+functions">multivariate Mittag-Leffler functions</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 07987126" data-ciurl="/ci/07987126" data-biburl="/bibtex/07987126.bib" data-amsurl="/amsrefs/07987126.bib" data-xmlurl="/xml/07987126.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07987126.pdf" title="Zbl 07987126 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s40314-024-03072-z" aria-label="DOI for “A general Volterra integral problem of Riemann-Liouville type”" title="10.1007/s40314-024-03072-z">DOI</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Al-Kandari, M.; Hanna, LA; Luchko, Y., On an extension of the Mikusiński type operational calculus for the Caputo fractional derivative, Integral Transform Spec Funct, 32, 9, 710-725, 2021; Al-Kandari, M.; Hanna, LA; Luchko, Y., On an extension of the Mikusiński type operational calculus for the Caputo fractional derivative, Integral Transform Spec Funct, 32, 9, 710-725, 2021 · <a href="/1491.44005" class="nowrap">Zbl 1491.44005</a> · <a href="https://doi.org/10.1080/10652469.2020.1833194" class="nowrap">doi:10.1080/10652469.2020.1833194</a></td> </tr><tr> <td>[2]</td> <td class="space">Calatayud, J.; Jornet, M.; Pinto, CMA, On the interpretation of Caputo fractional compartmental models, Chaos Solitons Fractals, 186, 2024; Calatayud, J.; Jornet, M.; Pinto, CMA, On the interpretation of Caputo fractional compartmental models, Chaos Solitons Fractals, 186, 2024 · <a href="https://doi.org/10.1016/j.chaos.2024.115263" class="nowrap">doi:10.1016/j.chaos.2024.115263</a></td> </tr><tr> <td>[3]</td> <td class="space">Cao Labora D (2025) An extension of the Cartwright-McMullen theorem in fractional calculus for the Stieltjes case. 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